Deep level Deligne--Lusztig induction for tamely ramified tori

TL;DR

This paper extends the construction of deep level Deligne--Lusztig representations to tamely ramified cases, showing their role in realizing supercuspidal representations of p-adic groups.

Contribution

It introduces a new construction of deep level Deligne--Lusztig varieties for tamely ramified tori, generalizing previous unramified cases.

Findings

Regular supercuspidals are compactly induced from deep level Deligne--Lusztig representations.

Irreducible supercuspidals are summands of cohomology of deep level Deligne--Lusztig varieties.

Construction applies under mild residue field assumptions.

Abstract

Deep level Deligne--Lusztig representations, which are natural analogues of classical Deligne--Lusztig representations, recently play an important role in geometrization of irreducible supercuspidals of $p$-adic groups. In this paper, we propose a construction of deep level Deligne--Lusztig varieties/representations in the tamely ramified case, extending previous constructions in the unramified case. As an application, under a mild assumption on the residue field, we show that each regular irreducible supercuspidal is the compact induction of a deep level Deligne--Lusztig representation, and generally, each irreducible supercuspidal is a direct summand of the compact induction of the cohomology of a deep level Deligne--Lusztig variety.